Name Class Date. This shows that A corresponds to Q. Therefore, A Q. This shows that BC corresponds to RS. Therefore, BC RS.

Transcription

1 Name lass ate eteaching ongruent igures iven, find corresponding parts using the names. rder matters. or example, or example, his shows that corresponds to. herefore,. his shows that corresponds to. herefore,. xercises ind corresponding parts using the order of the letters in the names. 1. Identify the remaining three pairs of corresponding angles and sides between and using the circle technique shown above. j, j, j, ngles: ides: 2. Which pair of corresponding sides is hardest to identify using this technique? nswers may vary. ample: and ind corresponding parts by redrawing figures. 3. he two congruent figures below at the left have been redrawn at the right. Why are the corresponding parts easier to identify in the drawing at the right? H H nswers may vary. ample: he drawing at the right shows figures in same orientation. 4. edraw the congruent polygons at the right in the same orientation. Identify all pairs of corresponding sides and angles. heck students work. j and j, j and j, j and j, j and j, j and j, and, and, and, and, and and all correspond. 5. N. Identify all pairs of congruent N sides and angles. j, j, j, j, N opyright by earson ducation, Inc., or its affiliates. ll ights eserved.

2 Name lass ate eteaching (continued) ongruent igures roblem iven, m = 30, and m = 65, what is m? How might you solve this problem? ketch both triangles, and put all the information on both diagrams m = 30; therefore, m = 30. How do you know? ecause and are corresponding parts of congruent triangles. 30 xercises Work through the exercises below to solve the problem above. 6. What angle in has the same measure as? What is the measure of that angle? dd the information to your sketch of. j; ou know the measures of two angles in. How can you find the measure of the third angle? nswers may vary. ample: Use riangle ngle-um hm. et sum of all three angles equal to What is m? How did you find your answer? 85; answers may vary. ample: mj = 180 ( ) efore writing a proof, add the information implied by each given statement to your sketch. hen use your sketch to help you with xercises dd the information implied by each given statement. 9. iven: and are right angles. mj = mj = 90, # and # 10. iven: and. is a parallelogram because it has opposite sides that are congruent. 11. iven:. } 12. an you conclude that using the given information above? If so, how? es; use the hird ngles hm. 13. How can you conclude that the third side of both triangles is congruent? he third side is shared by both triangles and is congruent by the efl. rop. of ongruence. opyright by earson ducation, Inc., or its affiliates. ll ights eserved.

3 Name lass ate eteaching riangle ongruence by and ou can prove that triangles are congruent using the two postulates below. ostulate 4-1: ide-ide-ide () ostulate If all three sides of a triangle are congruent to all three sides of another triangle, then those two triangles are congruent. If X,, and X, then X. In a triangle, the angle formed by any two sides is called the included angle for those sides. X ostulate 4-2: ide-ngle-ide () ostulate If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then those two triangles are congruent. If,, and, then. is included by and. is included by and. xercises 1. What other information do you need to prove by? ; by the eflexive roperty of It is given that j. hese are the included angles for the corresponding congruent sides. 2. What other information do you need to prove by? xplain. j; hese are the included angles between the corresponding congruent sides. 3. eveloping roof opy and complete the flow proof. iven:, iven iven Vertical are. j opyright by earson ducation, Inc., or its affiliates. ll ights eserved.

4 Name lass ate eteaching (continued) riangle ongruence by and Would you use or to prove the triangles congruent? If there is not enough information to prove the triangles congruent by or, write not enough information. xplain your answer X Not enough information; two pairs of corresponding Not enough Not enough information; only sides are congruent, but the information; you two pairs of corresponding congruent angles are not the need to know if sides are congruent. ou need included to know X or j. 7. iven:, is the 8. iven: HI H, H # I midpoint of N. rove: HI H rove: N N tatements: ; 2) is the midpoint of N; 3) ; 4) j; 5) ; easons: 1) iven; 2) iven; 3) ef. of midpoint; 4) Vert. ; 5) 9. carpenter is building a support for a bird feeder. He wants the triangles on either side of the vertical post to be congruent. He measures and finds that and that. What would he need to measure to prove that the triangles are congruent using? What would he need to measure to prove that they are congruent using? H tatements: 1) H; 2) H, H # I; 3) jh and jhi are rt. ; 4) jhi; 5) H; easons: 1) efl. rop.; 2) iven; 3) ef. of perpendicular; 4) ll rt. ; 5) I or, he would need to determine if j; for, he would need to determine 10. n artist is drawing two triangles. he draws each so that two sides are 4 in. and 5 in. long and an angle is 55. re her triangles congruent? xplain. nswers may vary. ample: aybe; if both the 55 angles are between the 4-in. and 5-in. sides, then the triangles are congruent by. opyright by earson ducation, Inc., or its affiliates. ll ights eserved.

5 Name lass ate eteaching riangle ongruence by and roblem an the ostulate or the heorem be applied directly to prove the triangles congruent? H a. ecause and are right angles, they are congruent. by the eflexive roperty of, and it is given that. herefore, by the heorem. xercises Indicate congruences. 1. opy the top figure at the right. ark the figure with the angle congruence and side congruence symbols that you would need to prove the triangles congruent by the ostulate. 2. opy the second figure shown. ark the figure with the angle congruence and side congruence symbols that you would need to prove the triangles congruent by the heorem. b. It is given that H H and. ecause H and H are vertical angles, they are congruent. herefore, H H by the ostulate. 3. raw and mark two triangles that are congruent by either the ostulate or the heorem. heck students work. What additional information would you need to prove each pair of triangles congruent by the stated postulate or theorem? ostulate 5. heorem 6. ostulate j j, jvu j, j, or j 7. heorem 8. heorem 9. ostulate j j j X U V opyright by earson ducation, Inc., or its affiliates. ll ights eserved.

6 Name lass ate eteaching (continued) riangle ongruence by and 10. rovide the reason for each step in the two-column proof. iven: X } VW, U VU, XU WVU, UWV is a right angle. rove: UX VUW tatements 1) UWV is a right angle. 2) VW # XW 3) X } VW 4) X # XW 5) UX is a right angle. 6) UWV UX 7) U VU 8) XU WVU 9) UX VUW easons 1)? iven 2)? efinition of perpendicular lines 3)? iven 4)? erpendicular ransversal heorem 5)? efinition of perpendicular lines 6)? ll right angles are congruent. 7)? 8)? 9)? iven iven X heorem U V W 11. Write a paragraph proof. iven: WX } ; W } X rove: WX W W It is given that WX } and W } X, so jw and jw, by the lternate Interior hm. X W by the eflexive roperty o, by ost. W. 12. eveloping roof omplete the proof by filling in the blanks. iven:, rove: roof: and 1 2 are given. by?. 4 2 o, by?. efl. rop. of ongruence Write a paragraph proof. 1 6 iven: 1 6, 3 4, 2 5 rove: N 4 N 3 j4 is given. herefore, mj3 mj4, by def. js. ecause j2 and j3 are linear pairs, and j4 and j5 are linear pairs, the pairs of angles are suppl. herefore, j5 by the ongruent uppl. hm. j6 are given, N, by the hm. opyright by earson ducation, Inc., or its affiliates. ll ights eserved.

7 Name lass ate eteaching Using orresponding arts of ongruent riangles If you can show that two triangles are congruent, then you can show that all the corresponding angles and sides of the triangles are congruent. roblem iven: }, rove: In this case you know that }. forms a transversal and creates a pair of alternate interior angles, and. ou have two pairs of congruent angles, and. ecause you know that the shared side is congruent to itself, you can use to show that the triangles are congruent. hen use the fact that corresponding parts are congruent to show that. Here is the proof: tatements easons 1) } 1) iven 2) 2) lternate Interior ngles heorem 3) 3) iven 4) 4) eflexive roperty of ongruence 5) 5) 6) 6) xercises 1. Write a two-column proof. iven: N, N rove: N N tatements easons 1)? 1) iven 2) 2)? eflexive roperty 3)? 3)? 4) N 4)? opyright by earson ducation, Inc., or its affiliates. ll ights eserved.

8 Name lass ate eteaching (continued) Using orresponding arts of ongruent riangles 2. Write a two-column proof. iven: is a median and an altitude of. rove: bisects. tatements 1) is a median of. 2)? is the midpoint of. 4) is an altitude of. 5) # 6) and are right angles. 7)? j 10) 11)? bisects j. easons 1)? iven 2) efinition of median 3) efinition of midpoint 4)? iven 5)? efinition of altitude 6)? efinition of perpendicular 7) ll right angles are congruent. 8) eflexive roperty of ongruence 9) 10)? 11)? efinition of angle bisector 3. Write a two-column proof. iven: ; bisects. rove: tatements easons ; bisects j. 1) iven 2) j 2) ef. of j bis. 3) efl. rop. of ongruence 4) 5) 4. Write a two-column proof. iven: N bisects H, H rove: N HN H N tatements easons 1) N bisects jh, jh 1) iven 2) jhn 2) ef. of j bis. 3) N 3) efl. rop. of ongruence 4) HN 4) 5) HN 5) opyright by earson ducation, Inc., or its affiliates. ll ights eserved.

9 Name lass ate eteaching (continued) Isosceles and quilateral riangles roblem What is the value of x? 40 ecause x is the measure of an angle in an equilateral triangle, x = 60. roblem y x 2 2 What is the value of y? m + m + m = 180 here are 180 in a triangle y = 180 ubstitution roperty y = 50 ubtraction roperty of quality xercises omplete each statement. xplain why it is true. 1.? j; base angles of an isosceles triangle are congruent. 2.? j; the angles of an equilateral triangle are congruent. 3.? ; the sides of an equilateral triangle are congruent. etermine the measure of the indicated angle lgebra ind the value of x and y. 7. y x 35; ; 50 y x 9. easoning n exterior angle of an isosceles triangle has a measure 140. ind two possible sets of measures for the angles of the triangle. 40, 40, 100; 40, 70, 70 opyright by earson ducation, Inc., or its affiliates. ll ights eserved.

10 Name lass ate eteaching (continued) ongruence in ight riangles xercises etermine if the given triangles are congruent by the Hypotenuse-eg heorem. If so, write the triangle congruence statement U V N not N V V X easure the hypotenuse and length of the legs of the given triangles with a ruler to determine if the triangles are congruent. If so, write the triangle congruence statement. 5. HI I 7. xplain why N N. Use the Hypotenuse-eg heorem. ecause jn and jn are right angles, both triangles are right triangles. It is given that their hypotenuses are congruent. ecause they share a leg, one pair of corresponding legs is congruent. ll criteria are met for the triangles to be congruent by the Hypotenuse-eg heorem. N 8. Visualize and, where = and =. What else must be true about these two triangles to prove that the triangles are congruent using the Hypotenuse-eg heorem? Write a congruence statement. j and j are right angles, or j and j are right opyright by earson ducation, Inc., or its affiliates. ll ights eserved.

11 Name lass ate eteaching (continued) ongruence in verlapping riangles eparate and redraw the overlapping triangles. Identify the vertices. 1. and H 2. and N 3. and H N U H N ill in the blanks for the two-column proof. 4. iven:,, rove: tatements 1),, 1)? 2) 3), 3)? easons 4) iven 5) = 5)? 6) + =, + = 6)? 7) + = + 7)? 8)? = 8) ubtr. rop. of quality 9)? Use the plan to write a two-column proof. 5. iven: and are right angles,. rove: lan for roof: rove by. hen use and vertical angles to prove by. iven ef. eg. ddition ost. ubstitution roperty tatements: 1) j and j are rt. ; j; 2) j; ; ; 5) j; ; ; easons: 1) iven; 2) t. are congruent; 3) efl. rop. 4) ; 5) Vert. 6) ; 7) opyright by earson ducation, Inc., or its affiliates. ll ights eserved.

12 Name lass ate eteaching ongruence ransformations ecause rigid motions preserve distance and angle measure, the image of a rigid motion or composition of rigid motions is congruent to the preimage. ongruence can be defined by rigid motions as follows. wo figures are congruent if and only if there is a sequence of one or more rigid motions that map one figure onto the other. ecause rigid motions map figures to congruent figures, rigid motions and compositions of rigid motions are also called congruence transformations. If two figures are congruent, you can find a congruence transformation that maps one figure to the other. y roblem In the figure at the right, U. What is a congruence transformation that maps to U? U appears to have the same shape and orientation as, but rotated 90, so start by applying the rotation r (90, ) on the vertices of. r (90, ) () = (-4, 1), r (90, ) () = (-1, 4), r (90, ) (r) = (-2, 1) raph the image r (90, ) ( ). translation of 1 unit to the right and 5 units down maps the image to U. herefore, ( 61, -57 r (90, ) )( ) = U. U y x x xercises ind a congruence transformation that maps to. U 1. y 2. y x x nswers may vary. ample: ( x@axis *5, 0+ )( ) = nswers may vary. ample: (r (270, ) y@axis )( ) = opyright by earson ducation, Inc., or its affiliates. ll ights eserved.

13 Name lass ate eteaching (continued) ongruence ransformations If you can show that a congruence transformation exists from one figure to another, then you have shown that the figures are congruent. roblem Verify the ostulate by using a congruence transformation. iven:,, rove: tart by translating so that points and coincide. ecause you are given that, there is a rigid motion that maps onto by rotating about point so that and coincide. hus, there is a congruence transformation that maps to, so. xercises 3. Verify the ostulate for triangle congruence by using congruence transformations. iven: X, X, rove: X nswers may vary. ample: X, translate so coincides with X. hen reflect across X to complete the transformation. 4. Verify the ostulate for triangle congruence by using congruence transformations. X iven:,, rove: nswers may vary. ample: ranslate so that points and coincide. hen rotate about point until and coincide. opyright by earson ducation, Inc., or its affiliates. ll ights eserved.